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2.1. WHAT ARE POLYNOMIALS? 37
While characterisations 3 and 2 of a Gröbner base (theorem 13) can make sense
in either view, characterisations 4 and 1 (the only effective one) only make sense
in a distributed view. Conversely, while the abstract definitions of factorisation
and greatest common divisors (definition 25) make sense whatever the view, the
only known algorithms for computing them (algorithm 1 or the advanced ones
in chapter 4) are inherently recursive 15 .
2.1.5 Other representations
Sparse representations take up little space if the polynomial is sparse. But
shifting the origin from x = 0 to x = 1, say, will destroy this sparsity, as
might many other operations. The following example, adapted from [CGH + 03],
illustrates this. Let Φ(Y, T ) be
∃X 1 . . . ∃X n (X 1 = T + 1) ∧ (X 2 = X 2 1 ) ∧ · · · ∧ (X n = X 2 n−1) ∧ (Y = X 2 n). (2.4)
The technology described in section 3.5.2 will convert this to a polynomial equation
Ψ(Y, T ) : Y = (1 + T ) 2n . (2.5)
Dense or sparse representations have problems with this, in the sense that expression
(2.4) has length O(n), but expression (2.5) has length O(2 n ) or more.
A factored representation could handle the right-hand side, assuming that we
are not representing the equations as polynomial = 0. But changing the last
conjunct of Φ to (Y = (X n + 1) 2 ) changes Ψ to
Y =
(
1 + (1 + T ) 2n−1) 2
, (2.6)
whose factored representation now has length O(2 n ).
Factored representations display a certain amount of internal structure, but
at the cost of an expensive, and possibly data-expanding, process of addition.
Are there representations which do not have these ‘defects’? Yes, though they
may have other ‘defects’.
Expression tree This representation “solves” the cost of addition in the factored
representation, by storing addition as such, just as the factored
representation stored multiplication as such. Hence ( (x + 1) 3 − 1 ) 2
would
be legal, and represented as such. Equation (2.6) would also be stored
compactly provided exponentiation is stored as such, e.g. Z 2 requiring
one copy of Z, rather than two as in Z · Z. This system is not canonical,
or even normal: consider (x + 1)(x − 1) − (x 2 − 1). This would be
described by Moses [Mos71] as a “liberal” system, and generally comes
with some kind of expand command to convert to a canonical representation.
Assuming now that the leaf nodes are constants and variables,
15 At least for commutative polynomials. Factorisation of non-commutative polynomials is
best done in a distributed form.